Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

download Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

of 10

Transcript of Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    1/10

    Min ia tu r i za t ion

    of

    H v d r o w o c e s s i n a C at alv s t

    Tes t ing Sys tem s : Theory and Prac t i ce

    S.

    T.

    Sie

    Faculty of Chemical Technology an d M aterials Science, Delft Un iversity of Technology,

    2628 BL Delft, T he N etherlands

    Main factors limiting downscaling of fi ed -b ed integral reactors are discussed, which

    operate with a gas stream

    or

    with gas and liquid i n trickle flow, as used in hydroproc-

    essing

    of

    oil. Criteria developed for a sufjiciently close approach to plug flo w and

    for

    good contacting

    of

    the catalyst show th at they can be m et i n very small reactors, both

    for processes with reactants in the gas phase and for trickle-flow processes, provided

    that the catalyst bed is diluted with fine inert material in the latter case. Experimental

    tests show that microreactors with typical catalyst volumes

    of 5-10 mL

    can be used to

    obtain representative results that can very well match data from industrial reactors. It

    appears to be feasible to m iniaturize catalytic test reactors jkrthe r to “nanosca1e”reac-

    tors with as little as 0.2-0.4 mL

    of

    catalyst while maintaining th e results to be me an-

    ingful. Even though there are advantages as well as limitations, miniaturization can

    enhanc e safety and reduce manpower.

    Introduction

    Miniaturization of equipment has many advantages and can

    in some cases even revolutionize fields of technology. Ekam-

    ples of drama tic impacts of miniaturization are in the field of

    biochemistry where the advent

    of

    microscale chromato-

    graphic separations paper, thin layer, and capillary high-per-

    formance liquid chromatography HPLC)) and other mi-

    croscale analytical techniques have been crucial to progress

    in this field for the last decades, a nd in the field of electronic

    computing, where miniaturization is causing a revolution in

    information technology.

    In the industrial chemical laboratory, too, reducing the

    scale of experimentation without detracting from the value of

    the results can be quite important, although the conse-

    quences of miniaturization may not be as spectacular as in

    the cases just cited. The incentives behind downscaling are

    manifold, and may include the ones below:

    Che aper equipment to construct and install

    Fewer materials to consume, store, and dispose of

    Fewer demands

    on

    laboratory infrastructure, including

    Intrinsically safer

    May promote better, more precise, working practices.

    The present article deals with the topic

    of

    scale reduction

    in research on catalytic processes. Catalysis, being a molecu-

    lower utility requirements

    lar phenomenon, can in principle be studied on a very small

    scale. For example, a microgram sample of a compound hav-

    ing a molecular weight of 600 consists of as many as about

    10’’ molecules, which is more than sufficient for this sample

    to be truly representative.

    As

    a matter

    of

    fact, fundamental

    catalysis studies on single-crystal surfaces equivalent to one

    microgram of a practical catalyst often involve minute

    amounts of reactants, and these studies are made possible by

    the high sensitivity of modern analytical techniques such as

    mass spectrometry, gas chromatography, and spectroscopic

    techniques that are also based o n molecular phenom ena.

    In contrast to these fundamental studies, research and de-

    velopment studies on industrial catalytic processes are not

    only concerned with understanding the basic chemistry of

    these processes but also with relating the information gener-

    ated in the laboratory with industrial practice. Since the com -

    position of practical feedstocks such as petroleum fractions

    may be very complex, and in view of the complicated reaction

    networks involved, a fundamental kinetic approach is only

    possible in a limited number of cases, notwithstanding the

    possibilities offered by modern analytical techniques and

    powerful computers. Therefore, there is a need for the ability

    to simulate the performance of an industrial reactor in the

    laboratory,

    3498 December 1996

    Vol.

    42, No. 12 AIChE Journal

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    2/10

    Traditionally, the laboratory equivalent of an industrial re-

    actor as used in hydroprocesses meters in width and tens of

    meters in height) was a pilot plant of much smaller dimen-

    sions but still very large according to laboratory standards

    e.g., a catalyst bed a few centimeters wide and several me-

    ters high). The subject of the present article is the downscal-

    ing of this pilot plant to more convenient dimensions while

    retaining the validity of the results obtained. The article will

    focus on the downsizing of fixed-bed reactors, which are

    among th e types of reactors m ost widely used in the pe troleum

    industry. Fixed-bed reactors operated with a gaseous stream

    of reactant as well as with a com bination of gas and liquid,

    namely, trickle-flow reactors, will b e c onsidered.

    Downscaling Approach for Continuous-Flow

    Fixed-Bed Reactors

    For a fixed-bed reactor operated under steady conditions,

    the important scaling factors are the dimensions of the cata-

    lyst bed and the catalyst particles. To simulate an industrial

    reactor generally operated as an integral reactor achieving

    practical levels of conversion of the reactant, the laboratory

    reactor must also be operated as an integral reactor giving

    the same conversion. The approach generally taken is to op-

    erate th e laboratory reactor at the same space velocity as the

    industrial one. Since at a constant space velocity the linear

    velocity of the reactant stream is directly proportional to the

    bed length, a reduction of bed length will reduce the linear

    velocity and th e Reynolds numb er accordingly

    Re=

    d p

    *

    u/q,

    in which d p is the particle diameter, the superficial veloc-

    ity, and

    q

    the dynamic viscosity). In principle, this means that

    the hydrodynamics in the commercial reactor and the shorter

    laboratory reactor will be different.

    Hydrodynam ic factors play a crucial role in tha t they affect

    the reaction rates in an interactive way with the intrinsic

    chemical kinetics, so one cannot reduce the reactor length,

    or

    at least not by a large factor, without losing representative-

    ness of the laboratory reactor. The only dimension that can

    be reduced is the reactor width, down to a size where the

    wall effect begins to affect results to be discussed later).

    These considerations are m ore o r less implied in th e sizing of

    the traditional pilot plant for fixed-bed processes: reactor

    lengths of 5 to 10 m and reactor diameters of 4 o 10 cm.

    Fortunately, however, in most fixed-bed processes of prac-

    tical interest, in particular th e widely applied fixed-bed proc-

    esses in the petroleum industry, hydrodynamics play a less

    important role than chemical kinetics and intraparticle diffu-

    sion in determining conversions, In the general situation of

    slow or moderately fast reactions occurring over catalyst par -

    ticles with effectiveness factors not much below one, the dif-

    fusion paths within the catalyst particle are of the order of

    the particle radius, while the effective film thicknesses out-

    side the particle involved in gas-solid, liquid-solid, or

    gas-liquid mass transfer will b e substantially sm aller. This

    point is illustrated in Table 1, which gives the relative magni-

    tude of the various mass-transfer resistances in a typical case

    of trickle-flow hydrotreating. Un der these circumstances, hy-

    drodynamics are only

    of

    importance insofar as they deter-

    mine pressure drops, holdups in multiphase flow, and in cer-

    tain cases distribution of fluids over the reactor cross section

    and heat-transport phenomena.

    Table

    1.

    Relative Mass-Transfer Resistances Estimated for a

    Typical Case

    of

    Gas Oil Hydro desulfurization n Trickle

    Flow

    Resistance of Total

    Negligible

    2

    21

    From bulk gas to

    G/L

    interface

    From G/L interface to bulk liquid

    From bulk liquid to ext. cat. surface

    Intraparticle diffusion and reaction 77

    When the desired function of the laboratory reactor does

    not include being representative o n the last-mentione d points,

    but is restricted to representativeness in a chemical reaction

    sense only, considerable f urth er dow nscaling is possible.

    As

    a

    starting point, one may assume that a well-designed indus-

    trial fixed-bed reactor is a close approximation of an ideal

    integral rea ctor tha t fulfills the following criteria:

    1.

    All volume elements of the feed must spend the same

    time in the reactor so as to contribute equally to the overall

    conversion. In other words, the reactor should be a close ap-

    proximation of a plug-flow reactor, which is the type of reac-

    tor to be selected for reactions of a positive order.

    2.

    All

    parts of the catalyst bed must contribute maximally

    to the overall conversion, implying that all catalyst particles

    must be in good contact with the reactant stream.

    To be representative

    of

    an industrial reactor that comes

    close to this ideal, the labo ratory reactor must also fulfill these

    requirements. Although improper design may give rise to in-

    dustrial reactors that p erform m ore poorly, it does not make

    much sense to try to mimic a malfunctioning reactor on a

    small scale. Therefore, compliance with the preceding re-

    quirem ents for an ideal reactor can be used a s a guiding prin-

    ciple in downscaling

    of

    fixed-bed reactors, and in the follow-

    ing sections we will examine how far reactor dimensions can

    be reduced without violating these requirements.

    Limits in Downsizing

    of

    Fixed-bed Reactors

    Deviations from

    plug

    f tow

    The allowable devia-

    tion fro m ideal plug flow is determined by what is considered

    to be an acceptable difference in conversion rate achieved in

    the ideal and the real reactor, and is also a function of the

    ord er of the reaction and th e degree of conversion. How much

    the residence time distribution in a practical reactor resem-

    bles that in an ideal reactor is generally expressed by the

    PCclet number (Pe ), a dimensionless number that is numeri-

    cally about twice the equivalent number

    of

    mixers in series.

    Fo r a true plug-flow reactor both num bers a re infinitely large.

    Mea rs 1971) has derived a relation between Pe , the reac-

    tion order

    n ,

    and the conversion X based on the criterion

    that a good practical reactor should require no more than

    5 extra catalyst to achieve the sam e conversion as the ideal

    one. A similar relation has been proposed by Ge rm an 1988),

    which is based on the somewhat less stringent criterion that

    the te mp erature requirement for the sam e conversion should

    not be higher than theoretical by one degree Centigrade,

    which is considered to be t he accuracy

    of

    temperature defini-

    tion in practice. The latter relation, which will be used

    throughout this article, can be written as

    Allowable D eviation

    from

    Plug Flow.

    AIChE

    Journal December 1996 Vol. 42 NO. 2 3499

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    3/10

    in which

    L

    is the reactor length,

    u

    the superficial velocity,

    and

    D ,

    the overall axial dispersion, which is made up from

    contributions from axial molecular diffusion, convective dis-

    persion in the packing, and macroscopic velocity distributions

    in the reactor.

    This can be an important cause

    of spread in residence time in the reactor, particularly with

    gases of high diffusivity and a t low linear velocities low space

    velocities and shallow beds). For a rather demanding case,

    namely, hydrogen at ambient conditions at a gas hourly space

    velocity (GHSV) of 1,000, he minimum bed length needed to

    satisfy the criterion of Eq. 1 for a good reactor is shown in

    Figure

    1 . It

    follows that a microreactor with a bed length of

    10cm will in most cases be acceptable.

    The statistical variations

    in

    width, length, and direction of individual channels within the

    packing result in a dispersion that is characterized by the di-

    mensionless Bodenstein number Bo

    =

    d

    *

    u/D,, , hich is

    analogous to the Ptc let number but has the particle diameter

    d f as characteristic dimension.

    For axial dispersion in the packing, Eq. 1 can be written as

    Axial M olecular Diffusion.

    Axial Convectiue Dispersion.

    -

    - ----

    -

    n = 2

    n = I

    -

    -___

    -

    L / d f >

    8/Bo)*

    n

    * In {1/(1- X I ) .

    2)

    For flow through packed beds

    Bo

    is a function of Re, and

    a global correlation between these dimensionless numb ers has

    been established by Gierman

    on

    the basis of published data

    Gierman, 1988). According to this correlation Bo decreases

    with decreasing

    Re

    until an approximately constant value of

    about

    0.4

    is obtained in the range of interest for small labora-

    tory reactors. For the axial dispersion of the liquid in trickle

    flow, the constant value of Bo is an order of magnitude lower,

    or about 0.04. It is of interest to note that for trickle-flow

    processes as used in the petroleum industry the dispersion of

    the gas phase is generally unimportant, since the gas phase

    consists mostly of recycled hydrogen. Consequently, conver-

    sion of hydrogen per pass is relatively low and rather unim-

    portant, so that only the dispersion of the liquid-oil phase

    needs to be considered.

    L , c m

    12

    1

    1

    10

    1

    /I

    t

    85 90 95

    100

    CONVaKlON ,

    o A i

    I I 1 I I I

    I

    Figure

    1.

    Minimum bed length as a funct ion of conver-

    sion for a first- and second-order reactor as

    determined by molecular diffusion in flowing

    gas.

    Hydrogen at 1 bar and

    25°C;

    G H S V = 1,000 nL/ L.h); =

    0.4; I = 2.

    3500 December 1996

    L.cm

    (single phase

    f low)

    L,cm trickle flow)

    5oE

    1

    85 90 95 100

    CONVERSION

    ,

    Figure 2. Minimum bed length as a function of conver-

    sion for a first- and second-order reaction as

    determined by axial d ispersion n the packing.

    Figure 2 presents the minimum bed length calculated ac-

    cording to Eq.

    2

    with the values for B o mentioned earlier, for

    beds of particles of 2 mm diameter a typical size for fixed-bed

    catalysts). It can be seen that in the case

    of

    single-phase flow,

    for examp le, gas flow, a bed length

    of

    20 cm as can be accom-

    modated in a microreactor is adequate for not too demand-

    ing cases conversions not exceeding 90 ). However, for

    trickle-flow operation much longer beds are required. For in-

    stance, for

    90

    conversion in the case of a second-order re-

    action the minimum b ed len gth of 2 m is required, w hich is in

    the range of pilot-plant reactors. Hence, microreactors are

    clearly unsuitable for testing catalysts of particle size in the

    millimeter range in trickle flow, and pilot-plant reactors se em

    indicated.

    However, as can also be inferred from Figure 2, from the

    point of view of axial dispersion there should be little prob-

    lem in testing particles of 0.1 mm diameter in a 20-cm-long

    bed in a m icroreactor even for trickle flow. This is one of the

    motives behind the catalyst dilution technique to be dis-

    cussed later.

    Wall Effect and Other Causes of Transverse Macroscopic

    Ve-

    locity Profiles. In addition to the axial dispersion discussed

    earlier, which is an intrinsic characteristic

    of

    the packing and

    which finds its origin in the more or less random variation

    of

    velocity vectors on the scale of the particle diameter, more

    systematic differences in velocities may occur in the flow

    through the bed.

    A

    lateral velocity profile over the reactor

    diam ete r may be caused by nonuniform packing uneven

    compaction or segregation of particles in the case of a cata-

    lyst with a spread in particle size).

    Although the preceding lateral velocity profile may be

    avoided by careful filling of the reactor, the re is another, more

    inherent cause of systematic velocity differences, nam ely, the

    wall effect. The packing

    of

    particles cannot be the same very

    Vol. 42, No. 12

    AIChE

    Journal

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    4/10

    close to the wall of the reactor as the unperturbed packing

    farther away from it. Generally, there is a much greater

    voidage in the wall region than the average voidage in the

    interior of the reactor, as found both experimentally Bena ti

    and Brosilow, 1982) and in computer-simulated packings

    Zimmerman and Ng, 1986). Th e grea ter permeability in the

    wall zone, which tends to enhance the flow in combination

    with the flow retardation by the reactor wall, results in a ra-

    dial velocity profile with a maximum located one to a few

    particle diam eters away from the wall Fahien and Stankovic,

    1979; Vortm eyer a nd Sch uster, 1983).

    The wall effect gives rise to an appreciable spread in resi-

    dence time of the reactant. However, as the ratio between

    bed diameter and particle diameter increases, the flow

    through

    the

    wall zone will carry less weight, and at a suffi-

    ciently large ratio its effect becomes acceptably small. At a

    ratio higher than about 25 the wall effect no longer has a

    noticeable effect on the average properties of the bed, for

    example, average porosity and average permeability Chu and

    Ng, 1989).

    Figure 3 shows the relation between the minimum reactor

    diameter and the particle diameter with a value of

    25

    as a

    criterion for an acceptably small wall effect. This figure can

    also be read as the maximum particle diameter that can be

    applied in a reactor of a given diameter. According to this

    figure, a fixed-bed catalyst of a practical size in the millime-

    ter rang e can only be tested in reactors of several centime ters

    width if the effect of the wall is to be minimal. Hence, for

    testing catalysts of such a practical size, a pilot-plant reactor

    with a diameter of at least 4 cm seems indicated, and a mi-

    croreactor with a diameter of

    l

    crn or less is clearly unsuit-

    able. However, with respect to the wall effect there should be

    no problem in such a microreactor with much smaller parti-

    cles, for example,

    0.1

    mm diameter, and this is a further rea -

    son

    for applying the catalyst-bed dilution techn ique t o be dis-

    cussed later.

    dp,mm

    0 1 2 3 4 5 6 7 8

    Figure 3. Maximum part ic le size for a given b ed diam e-

    ter, as dictated by the cri terion for n egl igible

    wall effect.

    0

    m

    Reducing the EfSect of Lateral Flow Profiles by Lateral Difsu

    sion. Th e preced ing criterion for negligible wall effect is un-

    duly stringent in the specific case that there is a sufficiently

    fast exchange of mass between regions of low and high for-

    ward velocities. This applies specifically to fixed-bed reactors

    operated with gases, which have a relatively high molecular

    diffusivity. Fast lateral mass exchange reduces the spread in

    residence tim e caused by the lateral velocity profile, a nd the

    interaction between axial velocity differences and lateral ex-

    change results in a dispersion that can be described by an

    axial diffusivity known as the Taylor diffusivity.

    Th e length equivalent

    to

    one mixing stage associated with

    the T aylor diffusivity can be w ritten as

    in which u is the average interstitial velocity, R the bed ra-

    dius, Dradhe diffusivity in the radial direction; is a dimen-

    sionless number that characterizes the flow profile. Combin-

    ing Eqs.

    1

    and

    3,

    one ob tains

    in which E is the bed voidage.

    The axial dispersion in packings caused by the combined

    effects of lateral flow profiles and radial diffusion has been

    the subject of many studies in chromatography. Information

    obtained in these studies is relevant to the present problem

    since the requirements of an ideal packed chromatographic

    column, namely, minimal band broadening and adequate

    contacting of the solid phase, are identical to those of an

    ideal fixed-bed reactor. In studies with packed chromato-

    graphic columns it has been found that for columns with a

    low ratio of tube diam eter over particle diam eter D/dp <

    5 )

    the value of is abou t 0.04, which is abou t twice the value of

    1/48 for the parabolic Poiseuille profile

    of

    laminar flow in a

    tub e Sie and R ijnders, 1967). Using the value of 0.04 and

    assuming the mechanism of radial diffusion to be molecular

    diffusion through the bed, we have calculated the maximum

    bed diameters for which radial diffusion is effective in wiping

    out the effect of the wall. No te that the re is a maximum

    limit on the bed diameter, whereas in the absence of fast

    radial diffusion, there is a

    minimum

    diameter for a negligible

    wall effect.) From results presented in Figure

    4

    t can be seen

    that with gas as flowing medium there should be no problem

    in testing catalyst particles of a practical size in a microreac-

    tor of cm diameter, notwithstanding the fact that the diam-

    eter ratio may be smaller than 5 .

    On

    the other hand, the

    diffusivity in liquids is too low to rely o n lateral exchange for

    effective counteraction of the wall effect in microreactors,

    since the maximum bed diameter for this to happen is in the

    millimeter range Figure 4).

    Measurement of residence-time distribution in microreac-

    tors ope rated with gas have confirmed that as a consequence

    of fast radial diffusion the contribution of the wall effect to

    the residence-time distribution is a relatively minor one. D ata

    shown in Table

    2

    show that this contribution is negligible as

    comp ared t o axial molecular diffusion.

    AIChE

    Journal December

    1996

    Vol.

    42,

    No.

    12

    3501

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    5/10

    D.cm

    Figure 4. Maximum b ed diameters for effect ive cou nter-

    act ion of rad ial f low pro f i les by radial di ffu-

    sion.

    Beds w i th D / d p <

    5.

    Gas : Hydrogen,

    P = 1

    bar ;

    T =

    25°C;

    G H S V

    =

    1,000 nL/ L.h) . Liquid: n-Eicosane. T

    =

    300°C;

    WH SV weight hour ly space veloci ty)= kg/ kg.h). E =

    0.4,

    r = 2

    Catalyst contacting

    The second requirement of an ideal reactor, namely, that

    all catalyst particles are in good contact with the reactant

    stream, will in most cases

    of

    a single fluid phase be fulfilled if

    the first criterion

    of a sharp residence time distribution is

    met. H owever, in the case of two flowing phases a sh arp resi-

    dence-tim e distribution is

    no

    guara ntee for adequate contact-

    ing. In a trickle-bed reactor a situation may prevail where

    liquid flows preferentially through a certain part of the bed,

    while gas flows mainly through the other part. Such macro-

    scopic maldistribution of liquid can be observed in experi-

    ments with colored liquids, but can also be simulated with

    comp uter models Zimm erman a nd Ng, 1986). This maldistri-

    bution, often referred to as incomplete wetting, may be sus-

    tained if the force

    of

    gravity responsible for th e dow nflow of

    liquid is much larger than the frictional forces. If the latter

    forces predominate, the high pressure drop will force liquid

    to flow through every available interstitial channel, and con-

    sequently catalyst contacting will be good.

    Table 2. Measured Axial Dispersion i n a Microflow Reactor

    with Gas Flow*

    No. of

    Dax   Dmo,/T

    Packing Pe Mixers cm2/s)

    (cm2/s>

    Spheres 105 53 0.132 0.137

    d , =

    1.5

    mm,

    E = 0.40

    d = 1.5 mm

    L = 3 m m ,

    c =

    0.44

    Cylinders 97 49 0.143 0.151

    L

    =

    10.2

    cm,

    =

    0.8

    cm,

    V

    =

    5

    mL. Nitrogen,

    P

    =

    1 bar ,

    T

    =

    25T ,

    G H S V

    =

    450 nL/ L.h) . Trace r : He. Binary dif fus ion coeff icient

    of

    H e in

    N,

    =

    0.678 cmZ/s, r

    =

    2 assumed) .

    A criterion for adequate wetting or even irrigation of cata-

    lyst is the following one proposed by Gierman and Harmsen

    Gierman, 1988)

    in which

    qL

    is the dynamic viscosity of the liquid, pL the

    liquid density, and g the gravity constant. The dimensionless

    constant

    W

    is the wetting number that compares frictional

    and gravitational forces. The numerical value in the preced-

    ing equation has been determined on the basis of a large

    volume of experimental data , and its order of magnitude can

    be rationalized by simple considerations Sie, 1991).

    Figure

    5

    shows the minimum bed length as a function of

    particle diameter required to meet the wetting criterion just

    given. The figure may also be read as the maximum particle

    diameter that can be applied in a bed of a given length. It

    can b e seen that, fo r liquid viscosities in the rang e typical for

    petroleum fractions under hydroprocessing conditions, a min-

    imum length of several meters is required with particles hav-

    ing a dia me ter of 2 mm and larger. Hence, for testing of such

    catalysts unde r trickle-flow conditions, a relatively large pilot

    plant reactor is required a nd a m icroreactor can be ruled out.

    However, the figure also shows that with much smaller parti-

    cles, for example, 0.2-mm diameter and smaller, the required

    bed length for good wetting of th e catalyst is less than

    20

    cm.

    Hence, with such small particles microflow reactors become

    eligible for testing catalyst for trickle-flow processes. This

    constitutes a third argument for the catalyst dilution tech-

    nique to be discussed below.

    L

    . m

    / I

    rl

    I I L I I

    I

    i l l 1 1

    0.5 5

    10

    dp.mm

    Figure 5. Minimum bed length for part ic les of varying

    size, as dictated by cri terion for even irr iga-

    tion.

    LHS V l iquid hour ly space veloci ty)

    = 1

    L/ L.h).

    3502

    December 1996

    Vol. 42, No. 12

    AIChE Journal

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    6/10

    Catalyst Bed Dilutionwith Fine Particles of

    Inert

    Material

    From the foregoing discussion it transpired that for fixed-

    bed processes operated with gas flow, reactors as small as

    microreactors about I-cm diameter,

    5

    to 20-cm bed length)

    can be used for testing catalysts of realistic size (1- to 3-mm

    diameter). In contrast herewith, relatively large pilot-plant

    reactors several meters high, several centime ters dia me ter)

    are required for testing such catalysts for trickle-flow proc-

    esses.

    In principle, one has the option of crushing the industrial

    catalyst and testing it in the form of fine particles in small

    reactors. How ever, aside from the danger of affecting catalyst

    properties by the crushing and sieving operations, the data

    obtained with crushed catalyst may not be representative for

    industrial practice. If pore-diffusion limitation occurs with the

    practical catalyst, the crushed sample will exhibit a higher

    activity. But even when the main reaction is not or is only

    slightly limited by intraparticle diffusion, other reactions in

    complex reaction networks involving a multitude of different

    feed molecules may be more seriously limited by pore diffu-

    sion. Hence, changing the catalyst particle size can affect not

    only the apparent activity of the catalyst, but selectivity and

    deactivation behavior as well.

    The dilemma

    of

    wanting to test relatively large industrial

    catalyst particles and needing fine particles in small beds for

    the hydrodynamic reasons discussed earlier can be solved by

    surrounding the catalyst particles with small granules

    0.05-0.2-mm diameter) of an inert material. Thus, the hy-

    drodynamics of the flowing fluids will be mainly dictated by

    the packing of small inert particles, whereas the catalytic con-

    1ISp-

    l l S f

    ’*

    I I W H S V ,

    h

    Figure 6. Second order plot for hydrodesulfurization of

    a light gas

    oil

    in trickle flow over undiluted

    beds of different lengths.

    D =

    8

    cm. Catalys t : Co/Mo/Alumina, 3-mm ellets.

    AXChE

    Journal December 1996

    version behavior is that of the catalyst in the actual size. The

    improvement in plug-flow approac h

    of

    laboratory trickle-flow

    reactors by adopting this catalyst-bed dilution technique has

    been demonstrated in experiments with isotopically labeled

    oils und er typical hydroprocessing conditions Van Klinken

    and van Dongen,

    1980).

    Another beneficial effect

    of

    catalyst-

    bed dilution is that temperature homogeneity in the bed is

    improved and that the chances of temperature runaway are

    diminished in th e case of strongly exothermic reactions, esp e-

    cially when using inert diluents with high thermal conductiv-

    ity, such as silicon carbide or alundum.

    The technique of catalyst-bed dilution opens the way to

    testing of industrial fixed-bed catalysts in very small labora-

    tory reactors, even for trickle-flow processes. This is sup-

    ported by th e results of bed-length reduction in gas-oil hydro-

    treating experiments, shown in Figures

    6

    and

    7.

    From the

    data for an

    undiluted

    bed shown in Figure 6, it can be in-

    ferred that shortening of a pilot-plant reactor from 4.8-m

    length by a factor of 3 is accompanied by a significant deteri-

    oration of the apparent catalyst performance. This can be

    understood from the previous discussion, since the length of

    the shorter reactor

    1.6

    m) is below the minimal lengths dic-

    tated by acceptable axial dispersion and by catalyst wetting

    cf. Figures 2 and

    5).

    By contrast, reduction of the length of a

    diluted

    cataIyst bed by a factor

    of

    2, namely, from

    40

    cm to

    20

    cm, does not noticeably affect the test results, as can be in-

    ferred from Figure

    7.

    In addition to supporting the earlier

    thesis that extraparticle mass-transfer effects play a minor role

    only, these results indicate t hat testing of diluted catalysts for

    trickle-flow processes is feasible in microreactors, since the

    5

    Figure

    7.

    Second order plot for hydrodesulfurization of

    a

    gas

    oil in trickle flow over diluted beds of

    different lengths.

    D

    =

    2 cm. Catalyst: Co/Mo /Alumina, 1.5-mm cylindrical

    ex

    t rudates . Di luent : 0.2 m m SIC.

    Vol. 42, No. 12

    3503

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    7/10

    length of the shorter catalyst bed

    is

    in the range covered by

    microreactors. Further evidence for this conclusion will be

    presented below.

    Table 4. Comparison

    of

    Microflow and Commercial Results

    for

    Trickle-FlowHydrodesulfurizationof Gas Oil*

    Reac tor Microflow Commercial

    Representat iveness

    of

    Catalyst Test ing in

    Microf low Reactors

    Fixed-bed processes with reactants in the gas phase

    From the previous discussion on th e limiting dimensions

    of

    lab ora toy fixed-bed reactors it followed that for these proc-

    esses microreactors with a bed diameter of about

    1

    cm and a

    bed length between 5 and 20 cm should give results tha t com e

    close to those of an ideal reactor, except for the most de-

    manding cases conversions very close to 100 ). Such results

    should therefore be representative for an industrial process,

    assuming that the latter is carried out in a well-designed re-

    actor.

    Evidence to support this statem ent is presented in Table

    3,

    which compa res microflow test results o n light paraffin isom-

    erization a process with reactants in the gas phase) with data

    on the same catalyst and feedstock from a commercial unit.

    It can be seen that there is a quite satisfactory agreement

    between the two sets of data. It is

    of

    interest to note that the

    catalyst in the form of extrudates of 1.5-mm dia me ter has not

    been diluted.

    Trickle-flow processes

    The suitability of microflow reactors with diluted catalyst

    beds for testing of industrial catalysts used in trickle-flow

    processes such as hydroprocessing of petroleum fractions is

    demonstrated by the data of Table 4,which is a direct com-

    parison between microflow test results and commercial oper-

    ating data

    on

    hydrodesulfurization of the same gas oil over

    the same catalyst under comparable conditions. In the mi-

    croflow test the catalyst bed of 1.2-mm extrudates was di-

    luted with 0.05-mm particles of silicon carbide. T he close cor-

    respondence between the two sets of data implies that the

    data generated in the microreactor test can be considered to

    be m eaningful for industrial practice.

    Miniatur izat ionof Catalyst Test ing below th e Sc ale

    of Microreactors

    From the analysis of the limiting geometric factors, pre-

    sented in the earlier part of this article, it can be concluded

    that th e microreactor scale is not necessarily the sm allest scale

    for representative testing of fixed-bed catalysts.

    As

    follows

    Table

    3.

    Comparison

    of

    Microflow and Commercial Results

    for Isomerization of Light Naphtha over a Ptmordenite

    Catalyst (1.5-mm Extrudates)

    Rea ctor Microflow Commercial

    Catalyst charg e 2.1 g 15.9 t

    Isopentane in product, 64.2 63.3

    Isohexanes in produ ct, 81.9 84.8

    2,2 diMe Butan e in 17.4 19.8

    C , ,

    yield, wt. 97.8 97.2

    RON-0

    of

    produ ct calc.) 79.6 79.7

    of total

    C,

    o f

    total C,

    product,

    o f

    total C,

    Catalyst volum e

    20 mL

    diluted) 122 m3

    Sulfur

    in

    produ ct, wt. 0.075 0.082

    Desulfurization 95.4 95.0

    Second-order rate constant

    23.5

    20.9

    *Catalys t : Co/Mo/Alumina, 1.2-mm tr i lobe extrudates . D i luent : 0.05 rnm

    Sic . Feed: Middle Eas t heavy gas o i l , 1 .64

    wt.

    S.

    Operat ing condi -

    t ions WH SV , WA BT weighted average bed tempera ture) , hydrogen/oil ,

    par t ial pressures of hydrogen a nd hydroge n sulf ide at reactor inlet ) same

    in both cases .

    from the earlier analysis, it is in principle possible to reduce

    the reactor diameter of about

    1

    cm typical for a microreac-

    tor) down to a few millimeters, while keeping the bed length

    about the same. The lower limit is in fact determined by the

    requireme nt t hat th e catalyst particles have to be accom mo-

    dated in the reactor, for example, for 1.5-mm-dia extrudates

    the minimum inner diameter of the reactor may be about 2

    mm. This implies a reduction in reactor and catalyst volume

    by a factor

    of

    about

    25,

    that is, instead of a typical catalyst

    volume of

    5-10

    mL in microflow reactors, the catalyst vol-

    ume in the submicro- or “nanoflow” reactor am ounts to as

    little as 0.2 to 0.4 mL.

    Even on the latter small scale, meaningful test results can

    be obtained, as will be demonstrated below.

    Fixed-bed processes with reactants in the gas phase

    Table

    5

    presents comparative test data obtained in mi-

    croflow and nanoflow reactors for the isomerization of light

    paraffins over a catalyst shaped as extrudates of 1.5-mm di-

    ameter. The nanoflow test data match the microflow test data

    quite well. The latter test data have been shown to be repre-

    sentative for catalyst performance under industrial condi-

    tions, and by implication this should also be true for the

    nanoflow test results.

    Figures 8 and 9 compare microflow and nanoflow test re-

    sults on t he hydrodesulfurization of the same gas oil over a

    catalyst shaped a s 1.5-mm extrudates, d iluted with 0.05-mm

    particles of silicon carbide. It can b e seen tha t th ere is a good

    match between the data obtained with the two types of reac-

    tors, bo th with respect to desulfurization r ate Figure

    8)

    and

    the effect of tem per atu re on desulfurization Figure 9).

    Table

    5.

    Comparison of Microflow and Nanoflow Test

    Results for Pen tan epex ane Isomerization*

    Rea ctor Microflow Nanoflow

    Catalyst charge , g 5 0.2

    Isopentane

    in

    produ ct, 58.3 59.5

    of

    total pentanes

    of total hexanes

    product, of total hexanes

    Isohexanes in product, 80.7 80.2

    2,2 diMeButane

    in

    16.4 16.5

    C 5+ yield, wt.

    99.0 98.9

    RON-0 of produ ct calc.) 77.8 77.9

    *Catalys t : Pt /Mo rdeni te , 1.5-mm extrud ates undi luted) . Fe ed : n-pen -

    tane/n-hexane/CycloC, 61/34/5 wt.

    76 . T =

    260°C; P

    =

    25 bar ; hydro-

    gen/feed

    =

    1.25 molar ; WHSV

    =

    1.70.

    3504 December 1996 Vol. 42, No. 12 AIChE Journal

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    8/10

      /sxp- /s f

    .8

    2.6

    2l4

    2.2

    4 1

    -

    -

    -

    -

    -

    -

    -

    -

    OMICROFLOW

    NANOFLOW

    I

    I

    I I

    0.1

    0.2 a3

    l /WHSV,

    h

    Figure

    8.

    Microf low vs. nanof low test resul ts for hydro -

    desulfu rizat ion of gas oil in tr ickle f low; effect

    of s pace velocity.

    Catalyst: Co/Mo/Alumina, 1.5-mm cylindrical extrudates.

    Di luent : 0.05

    mm

    S i c .

    x = 0.84.

    Since it has been shown that microflow test data can be

    representative for industrial operation, it follows from the

    preceding comparison that nanoflow test data, too, can be

    meaningful for industrial practice. This statement is substan-

    tiated by Table 6, where such test data on hydrodesulfuriza-

    tion of a gas oil over a catalyst shaped as 1.5-mm extrudates

    are directly compared with data from a commercial hydro-

    desulfurization unit.

    Potential errors in catalyst sampling

    Proper sampling from a catalyst batch composed of non-

    identical catalyst particles is obviously important for obtain-

    ing meaningful results. To obtain representative samples

    of

    such an inhomogeneous catalyst batch for testing in labora-

    tory reactors ranging from pilot plants to microreactors, the

    well-known procedures for sampling

    of

    solid granular mate-

    rial have to be adhered to. However, with a catalyst sample

    as small as that to be tested in nanoflow reactors, an addi-

    tional problem may arise due to the small number of parti-

    cles.

    A

    catalyst sample of 0.3 g, consisting of 1.5-mm-dia. and

    4-mm-long extrudates will typically consist of about

    40

    parti-

    cles, and at this small number the law of statistics has to be

    considered, Figure 10presents the calculated standard devia-

    tions for a random sampling from a mixture of two catalysts

    A and B ) that differ in activity by

    a

    factor k,/k,. Assuming

    that a standard deviation below 5 is acceptable, it can be

    seen that there is no problem when the sample size consists

    AIChE Journal December 1996

    l n

    301

     

    1.58

    1.60 1.62

    1.64

    10001T.

    K-1

    Figure

    9.

    Microf lo w vs. nanof iow test results for hydro-

    desulfu rizat ion of g as oi l in tr ickle f low; effect

    of

    temperature.

    Catalys t and di luent as s t a t ed und er Figure 8.

    of 10,000 particles, as long as the sampling is nonselective.

    However, at a sample size corresponding with

    20

    particles,

    the standard deviation may be above

    5

    when the activity

    ratio is larger than a bout 1.5 and when the less active catalyst

    constitutes a major proportion of the mixture. If the less ac-

    tive catalyst is a minor component of the mix,

    for

    example,

    lo ,

    the standard deviation remains below the set limit of

    5 , even at a very large activity ratio.

    The latter situation may occur in practice because a batch

    of catalyst can be contaminated with a small amount of less

    active or even inert material. In such a situation, the limited

    number of particles

    in

    the test sample does not necessarily

    pose a problem, as long as the sampling is done in a nonse-

    lective manner.

    Table

    6.

    Comparison of Nanoflow Test Results with Data of

    Commercial Operation on Hydrodesulfurizationof Gas Oil*

    Rea ctor Nanoflow Commercial

    Catalyst charge 0.228g + diluent) 33.5 t

    Sulfur in product, wt. 0.155

    0.144

    Desulfurization

    80.5 81.9

    2.1-Order rate constant 22 24

    *Catalys t : Co/Mo /Alumina, 1.5-mm t r i lobe extrudates . Di luent : 0.05 m m

    S i c . F eed : G as o i l

    162-415 C, 0.794

    wt.

    S.

    Condit ions

    WHSV,

    WABT, hydrogen/oi l and par t ial pressures of hydrogen and hydrogen

    sulf ide at reactor

    inlet)

    s am e

    for

    t he two cases.

    Vol.

    42, No. 12 3505

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    9/10

    STANDARD DEVlATlON

    IN 6

    2l

    1 1.2 1.4 1.6 1.8 2.o

    Figure 10. Sampl ing errors f rom a catalys t mix ture

    composed of two com ponents hav ing d i f fer-

    ent activit ies, for samples cons is t ing of N

    part ic les.

    UI

    kA kB

    How ever, when ther e is a large activity difference and when

    the less active material is abundant the statistical error may

    be unacceptably large for small samples. Fortunately, this is

    not a situation of much practical interest since such an inho-

    mogeneous catalyst is obviously not an optimal one and there

    is hardly a reason to try and assess the perform ance

    of

    such a

    poor batch of catalyst with great accuracy.

    Implementation and Consequences

    of

    Small-Scale

    Catalyst Testing

    The increased possibilities of catalyst testing on a smaller

    scale tha n considered feasible in the past, and the advan tages

    attached to experiments on a small scale, have led to a grad-

    ual replacemen t of the traditional large pilot plants by smaller

    units. This can be seen from Table 7, which compares the

    distributions of reactor units of varying size in a n oil-process

    R & D laboratory in 1970 and 1987. It can be seen that dur-

    ing the intervening period bench-scale and microflow units

    have taken over many of the tasks formerly reserved for larger

    pilot plants. In more recent years, microflow tests are being

    widely applied, not only in exploratory studies where in the

    screening of new catalysts and unconventional feedstocks the

    small sample sizes are very importan t, but also in the process

    research and development studies on more established proc-

    esses. In addition to the previously cited processes, namely,

    paraffin isomerization and hydrotreating of distillates, mi-

    Table

    7.

    Distribution

    of

    Number

    of

    Reactor Units*

    Year 1970 1987

    Microflow z

    0.01

    L) 13 50

    Bench Scale

    = 0.1

    L)

    20

    38

    Pilot Plant > 1 L) 67 12

    Unattended

    units

    13 94

    *At

    the

    oil process R &

    D

    Department of the Koninklijke/Shell Labora-

    torium, Amsterdam in of total).

    3506

    December

    1996

    croflow reactors play an important role in research and de-

    velopment on most hydrocarbon-conversion processes, in-

    cluding catalytic reformin g Sie and Blau who ff, 19911, hydro-

    cracking Van D ijk et al., 1991), and even hydrodemetalliza-

    tion and hydroconversion of residual oils Oelde rik et al.,

    1989). Th e use of m icroflow reactors is no t restricted to sim-

    ple once-through experiments in isothermally operated reac-

    tors, since on this scale accurate simulation

    of

    adiabatic oper-

    ation h as proven feasible Sie and B lauwh off, 19911, as well

    as simulation of a hydrocracker with on-line fractionation and

    recycle of fractionator bottoms Van Dijk et al., 1991).

    Driving forces behind this trend have been the improved

    research a nd dev elopment efficiency resulting from savings

    in

    costs, materials, and personnel, as well as the enhancement

    of safety and environm ental friendliness of the operation. T he

    savings in materials in moving from pilot plant to microreac-

    tor testing can be illustrated by comparing the feed require-

    ments of these two reactor units for a typical case of gas-oil

    hydrotreating: whereas the pilot plant consumes monthly

    amounts of oil and hydrogen gas that have to be supplied on

    a periodic basis by tank cars and tube trailers, the microflow

    needs in a similar period can be covered by a jerry can of oil

    and a few gas bottles. In th e pilot-plant case, the storage and

    disposal of the large amount of unused products may also

    constitute a logistic problem with environmental aspects.

    The enhanced intrinsic safety resulting from miniaturiza-

    tion may be illustrated by the m aximum hazards of a nanoflow

    reactor. When working with flammable gases or liquids, flow

    rates are

    so

    small that they can only sustain a flame com para-

    ble in size to that of a small candle. Working with synthesis

    gas, the small flow rate may give rise to an emission of toxic

    carbon monoxide that is of the order

    of

    that from a burning

    cigarette. The maximum release of compressed gas from the

    reactor inventory upon mechanical failure would be less than

    from a pu nctu red bicycle tire and if this gas is explosiv e, its

    detonating power can be compared with that of a small fire-

    cracker. Hence, many of the safety systems mandatory in the

    case of larger reactors fire detectors, flammable a nd toxic-gas

    detectors, special ventilation, etc.) can be simplified or even

    dispensed with.

    Th e higher intrinsic safety of smaller reactor units makes it

    easier to automate these units and to allow them to operate

    around the clock withput human attention. The increase in

    automation, which has accompanied the size reduction of the

    reactor units in the period 1970-1987, is clearly shown in

    Table 7, where the percentage of unattended units is seen to

    increase from 13 to 94.

    The increase in research productivity resulting from cata-

    lyst testing on a smaller scale is illustrated by Figure 11. It

    can be inferred tha t the more m odern, autom ated microreac-

    tor unit operating without human attention requires an ord er

    of magnitude less manpower than the traditional large pilot

    plant it has replaced.

    Concluding Remarks

    It has been shown that microflow and even smaller reac-

    tors can be used not only in fundamental kinetic studies with

    powdered catalysts, but also for testing practical fixed-bed

    hydroprocessing catalysts under industrially relevant condi-

    tions, that is, with catalysts in their actual shape and size,

    Vol. 42, No. 12

    AIChE Journal

  • 8/18/2019 Miniaturization of Hvdrowocessina Catalvst Testing System -Sie-1996-AIChE_Journal

    10/10

    MANHOUR PER REACTOR HOUR

    21

    M

    ICROFLOW

    automated 1

    \

    \

    1960 I910 1980 1990

    Moo

    YEAR

    Figure 11. Manpower needs of different reactor units as

    a function of the approximate introduction

    year for general use in oil process research.

    with practical feedstocks, and at conditions

    of

    space velocity,

    temperature, and pressure that are comparable to those in

    practice, yielding data that can match those of commercial

    operation. This applies not only to fixed beds operated with a

    stream of gas, but equally to fixed beds operated with liquid

    and gas as trickle-bed reactors, provided the catalyst bed is

    diluted with fine inert material. This experimentally verified

    conclusion

    is

    supported by a theoretical analysis

    of

    the fac-

    tors that limit the downscaling possibility of fixed-bed reac-

    tors.

    These small reactors can now be used for generating cat-

    alytic performance data that some 25 years ago had to be

    done in large fixed-bed pilot plants, at only a fraction of the

    costs, materials usage, and personnel need associated with

    the operation of the large pilot plants. Hence, for many cat-

    alytic process studies there is no nee d for the traditional large

    pilot plant any more. Nevertheless, cases will remain where

    experiments on a large scale cannot be avoided, for example,

    when large quantities

    of

    a new product are needed for mar-

    ket development in anticipation of commercial production. A

    large pilot plant is also needed when hydrodynamic factors

    play a crucial role in determining t he chem ical conversions of

    a catalytic process or when m echanical/physical features, such

    as fouling of the bed by particulate matter, are important.

    Another reason for a large and complex integrated pilot plant

    is when the process configuration is complex and involves

    separation steps and recycles of streams that cannot be

    miniaturized in a representative way.

    For the evolution in catalyst testing systems, where minia-

    turization is increasingly offering capabilities at lower costs

    but cannot make large pilot plants totally redundant, an anal-

    ogy

    can be drawn with the evolution in computing: while mi-

    crocomputers are finding an ever increasing role, it is still

    necessary to make use of mainframe and even supercomput-

    ers in special applications.

    Acknowledgment

    The author gratefully acknowledges the valuable contributions to

    this work of his former coworkers at the Koninklijkefihell-Labora-

    torium Amsterdam, where most of the work was carried out. He is

    particularly indebted to Messrs J. Glezer and

    A.

    F. de Vries for their

    role in the development of the nanoflow equipment and

    in

    the exe-

    cution of the comparative tests.

    Notation

    Dmol

    molecular diffusivity,

    m2.s-’

    or cm2.s-1

    E

    =activation energy, kJ.mol-’

    k

    = reaction-rate constant

    P =pressure, bar

    Sf

    =sulfur content in feed, wt.

    S p

    =sulfur content in product, wt.

    T =temperature, degrees Centigrade

    V

    =catalyst bed volume, m3 or L

    =kinematic viscosity, m2.s-l

    T = ortuosity factor

    Literature

    Cited

    Benati, R. F., and C. B. Brosilow, “Void Fraction Distribution in

    Beds of Spheres,” AZChE J. 8, 359 (1982).

    Chu, C. F., and K. M. Ng, “Flow in Packed Tubes with a Small Tube

    to Diameter Ratio,”AIChE J. 35,148

    (1989).

    Fahien, R.

    W.,

    and

    I.

    M. Stankovic,

    “An

    Equation for the Velocity

    Profile in Packed Columns,”Chern. Eng. Sci., 34, 1350 (1979).

    Gierman, H., “Design of Laboratory Hydrotreating Reactors. Scaling

    Down

    of

    Trickle-Flow Reactors,”

    Appl. Catal. ,

    43,

    277 (1988).

    Mears,

    D.

    E., “The Role of Axial Dispersion in Trickle-Flow Labo-

    ratory Reactors,” Chem. Eng. Sci., 26, 1361 (1971).

    Oelderik, J. M.,

    S.

    T. Sie, and D. Bode, “Progress in the Upgrading

    of Petroleum Residue,” Appl. CatuL,

    47,

    (1989).

    Sie, S. T., “Scale Effects in Laboratory and Pilot-Plant Reactors for

    Trickle-Flow Processes,”

    Rev. Znst. Fr.

    Pit . , 46,

    SO1 (1991).

    Sie,

    S.

    T., and G. W. A. Rijnders, “Band-Broadening in Packed

    Chromatographic Columns,”

    Anal.

    Chim.

    A c f a ,

    38,

    3 (1967).

    Sie,

    S.

    T., and P. M. Blauwhoff, “Laboratory Equipment and Proce-

    dures for Evaluation of Catalysts in Catalytic Reforming,” Cafal.

    Today, 11, 103 (1991).

    Van Dijk,

    A., A.

    F. de Vries, J.

    A.

    R.

    van

    Veen,

    W . H. J.

    Stork, and

    P. M. M. Blauwhoff, “Evaluation of Hydrocracking Catalysts in

    Recycle Tests,” Cafal. Today, 11, 129 (1991).

    Van Klinken,

    J.,

    and R.

    H. van

    Dongen, “Catalyst Dilution for Im-

    proved Performance of Laboratoly Trickle Flow Reactors,” Chem.

    Eng. Sci., 35,59 (1980).

    Vortmeyer, D., and

    J

    Schuster, “Evaluation of Steady Flow Profiles

    in Rectangular and Circular Packed Beds by a Variational Method,”

    Chem.

    Eng.

    Sci. 38, 1691 (1983).

    Zimmerman, S. P., and K. M. Ng, “Liquid Distribution in Trickling

    Flow Trickle-Bed Reactors,”

    Chern. Eng. Sci., 41,

    861 (1986).

    Manuscript received

    Mar.

    19, 1996, and revision received

    May

    31, 1996.

    AIChE Journal

    December

    1996 Vol.

    42,

    No. 12

    3507